Mathematics

Getting to yes

A quick problem solving puzzle from the New York Times.

I got it right.  Click the button to read me talking about how I got there, but don’t do it till you’ve tried the puzzle yourself. Please.

I

5-1, 4-1, 3-1, 2-1

Over and over again Thursday night and Friday morning, my Twitter and Facebook streams lit up with the same, repeated message. The Florida basketball team had just eeked out a win against Brigham Young University in the Sweet Sixteen, and again and again I was informed, “The Florida Gators have NEVER lost in the Elite Eight.”

If I’d only seen it once or twice, I’d have probably just rolled my eyes, or even found it mildly amusing. But it wasn’t once or twice. I was bombarded with that one little trivium for almost 24 hours, and so I ended up going through a progressive series of reactions:

1. That’s a truly useless little statistic.

2. We’re not exactly Kentucky or Kansas or a school in the Research Triangle. (Though by the time Billy Donovan retires, we might be.) We’ve been to four Final Fours, so if we’re undefeated in the Elite Eight, that means we’ve been to (quick arithmetic on fingers) four Elite Eights. Maybe the reason we’ve never lost one is because we haven’t had terribly many opportunities to lose one.

3. Well, now the math thoughts are kicking in. Let’s see.

I’m pretty sure the Gators’s first Sweet Sixteen was the 1994 team who went to our first Final Four.* Which means I should be able to go through every UF trip to the Sixteen in my head.

The next time we were in the regional semi-finals was 1999, when we lost to Gonzaga in the midst of their first great Cinderella run. Since then, we’ve been to three more, all three of which resulted in a trip to the Final Four. So counting the win over BYU, that makes us 5-1 in the Sweet Sixteen. And after the game against Butler, we’ll either be 5-0 or 4-1 in the Elite Eight.

And then we come to the Final Fours. In 1994 we lost to Duke; in 1999, 2006 and 2007 we progressed to the national championship game. In our three national championship games, we lost to Michigan State in 1999, beat UCLA in 2006 and beat Ohio State in 2007.

So let’s see. 5-1. 5-0 or 4-1. 3-1. 2-1.

Well then, clearly it’s time to lose in the Elite Eight, for the symmetry.

Now, that whole thought process? I thought it to myself rather glibly, because I actually expected the Gators to win. (By contrast to the previous two rounds, where I’d gone in with a pessimistic feeling that we were going to end up being upset by, respectively, UCLA or BYU.) But when we ultimately went down in overtime, I silently thought at all the people who’d been so excited by our previously immaculate Elite Eight record, “Told you. Silently. In my head.”

It was pretty disappointing to see Florida lose that game, so to compound my misery, I hopped onto ESPN.com to see how badly I was doing in my bracket league. I’d been tied for last place after the Round of 64 and in second-last after the Round of 32. I was vaguely hopeful I’d worked my way up to third-last, or morbidly hopeful I’d managed to tie up dead last place.

Much to my surprise, I discovered that not only had I locked up second place, but that I was actually in the 97th percentile for ESPN’s entire bracket competition. And the remaining three of my Final Four teams were all in the three remaining Elite Eight games–and were in fact the higher-seeded team in each of those three games. I was looking forward to having successfully predicted three of the Final Four.

Well, thanks to VCU and Kentucky, it turns out I only predicted one team in the Final Four–UConn. But I’m still in ESPN’s 96th percentile, and I can presumably still improve on that position, since I have UConn winning the national championship. I can’t get to first place in our group, though, since my sister Claire also has UConn winning it all. But I still have the satisfaction that Claire and I were the only two people in our group to predict Kentucky beating Ohio State in the Sweet Sixteen.

Obviously, I’ll be cheering for UConn to beat Kentucky in the Final Four. And I’ll be cheering for VCU to beat Butler, both because I live in Virginia and because Shaka Smart was an assistant coach on the Gator basketball team that won consecutive national championships four years ago. If we end up with a UConn-VCU national championship game, I don’t know who I’ll be pulling for. More likely it’ll be UConn–a childhood spent in large part in Connecticut will do that to a person.

I

*When I looked it up later, I was wrong about that part. The Gators had previously been to the Sweet Sixteen in their first ever NCAA Tournament, in 1987. As a 6-seed, they beat 3-seed Purdue in the second round and lost to 2-seed Syracuse in the Sixteen. So we’re actually 6-1 in the regional semi-finals. Whatevs. Still a good point.

Snip snip buzz

Leo Jones (Reggie Yates)Okay, I don’t get it.

I go into a salon in Florida, and I tell them what I want–an inch long on top, half an inch on the sides. And I walk out of there with an inch of hair on top and half an inch on the sides.

But in Maryland or Virginia, I say an inch on top and half an inch on the sides. And always, I end up with the same result–two or even three inches long on top, and an inch and a half on the sides.

Do Floridians really have that much better of a handle on Imperial measurements than Marylanders and Virginians? As a product of the Florida public school system, frankly I find that rather hard to believe. But whatever the reason, I’m getting rather tired of having to time my haircuts so that they coincide with trips to Florida.

I

Inheritance
Words yesterday: 1089
Words total: 51,804

Time spent writing: Two hours (1.30-2.30, 3pm-4pm)
Reason for stopping: Lisa got home; Boy got up from his nap, quota
Food: Pasta topped with chili and Italian cheeses. God that’s the best lunch I’ve made myself in a while.

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Oh, and like Pink Floyd points out, also Thought Control

Victoria Waterfield (Deborah Watling)I’ve been thinking quite a bit lately about my beliefs, and have been kicking around the idea of trying to tie them together into some sort of cogent social philosophy, so that I’ll already have some groundwork completed when I start constructing my programme of reforms after the coming, inevitable Revolution sweeps me to power. (Note that I call it a social philosophy–as in “a philosophy of society”–and not a political philosophy.)

I’m also trying to decide if it would be at all beneficial to try to formulate a statement of my philosophy (a manifesto, some might call it), either to be written here or somewhere more private. Such a statement would probably be a pretty broad, longterm project, but right now I’m just going to be talking about the issue on which I usually feel the strongest in politics: public education.

This post might mark a bit of a departure from this blog’s usual tone. (Or, on the other hand, perhaps you’ll find it to be intellectual, introspective, condescending, grammatically correct, sarcastic and smugly elitist, and therefore exactly typical of what you usually associate with what I write here.) But if it’s a bit drier than you’re used to, I hope it’s also a bit more thought-provoking.

By education, I mean specifically secondary education, both because it’s (to me) the most important phase of education, and because I think it’s the phase we’re most prone to get wrong–because it’s the least straightforward. Primary education is basically about socialising children so that they’re capable of conducting themselves in groups, and about laying the academic basics that they’re going to be building their education on once they get to middle and high school; and post-secondary education (why don’t we call it tertiary education?) is something that gets tailored to each individual’s circumstance. But secondary education is where we’re supposed to be turning children into young men and women capable of functioning in our society.

We teach students five core subjects in secondary school: Math, Science, English, History and Foreign Language. But too often, I think, we think of these subjects as–well, as nouns, is the best way I can think of to put it. As set bodies of knowledge, to be learnt and memorised by the student, and for that knowledge to then be regurgitated in tests to prove that he or she has learnt it.

Now, this body-of-knowledge aspect of each subject is certainly important. I’d definitely expect someone who’s taken three or four years of Spanish to be capable of emailing hotels in Patagonia so that they can find one with plumbing in every room or which serves breakfast in the mornings. And I’d expect any high school graduate to be able to tell me what three things are defined by any group of three nonlinear points, how you mix acid and water, what we know about Shakespeare’s characters based on whether they speak in iambic pentameter or in blank verse, and why Great Britain, after spending three hundred years allying with the German states to continually frustrate French expansionism in Central Europe, suddenly made a volte face after 1871 and spent the ensuing century instead siding with France against the Germans.

But there’s something else that each of these subjects should be imparting to our children–something that most people overlook, but that is to me far more important than the conventional notion of the classroom being a place to learn hard facts. If the conventional idea of each subject is its “nounlike” aspect–the body of knowledge that just is, sitting there waiting to be learnt–then this other aspect is the subject as verb: each of these five core subjects is also a skill, a way of approaching the world and of seeing it from a different perspective. It’s vital both for our children and for the society they’ll inherit that they pick up these skills during their adolescence.

Maths. (Or math to my American friends.) This is probably the easiest to see as a skill rather than a book of facts–none of us have a problem with the concept of learning to “do maths.” Mathematics teaches logical reasoning, induction and a whole host of other critical thinking concepts.

Science. Fundamentally, all science begins with observation of natural phenomena. From those observations scientists make generalisations about how they think the world works–hypotheses–and then they test those generalisations and see if their test results are consistent with what their hypotheses predict. And if they’re not consistent, then they throw out the hypothesis, not the test result. What I’m describing is, of course, the Scientific Method.

And that’s the single most important thing we can hope for a student to take out of a Science classroom–the idea that you don’t know something, truly know it, until you’ve seen it for yourself, and examined it enough to test it and poke holes in it.

Which is of course why introducing religion, in the form of intelligent design, into the Science classroom is a blow aimed at the fundamental principle on which all rational scientific discovery is built. It’s an attempt to base the curriculum around an article of unobservable, untestable religious faith (that evolution is guided by a Higher Power)–and faith, being a belief that we hold regardless of the absence of independently verifiable evidence (or even despite evidence to the contrary), is absolutely anathaemic to science by definition.

(Please note that I’m not saying it’s impossible for a person to be both faithful and rational. It’s just that they deal with totally different areas. Faith is there for the why, while science is there for the how–though both sides of the debate sometimes forget that distinction. What I think a lot of the faithful–the sort of faithful, at least, who think intelligent design should be taught in our schools–often find so threatening is that when people are able to better understand the how through science, many of them no longer seem to need the why that faith is there to provide.)

English. By high school, English has become a literature class, not a grammar class. It’s (hopefully) where our children learn to look at subtext–to know that the writer (or the speaker) can be creating an atmosphere, and manipulating our reaction, beneath the surface of their words. Anyone who’s colour-marked should know what I’m talking about. It’s where they learn to pick the text apart–to understand even the things that aren’t being said. It’s where they learn how language itself can be used to manipulate their perception, so that they understand what’s being done when politicians refer to President Bush’s immigration reform proposal as “amnesty”, or when the commercials I’ve recently heard airing on the radio refer to an American withdrawal from Iraq as “surrender”.

Foreign Language. One of the things I often find frustrating about living in a foreign country is the way people just assume that everyone else in the world thinks like them. It’s also, I think, at the root of a lot of the ill will the United States seems to have generated for itself through its foreign policy over the last five years–the preconception that, given the same set of circumstances, most of the peoples would react the same way Americans would in a given situation.

Learning a second language is a great way to show people that that’s not true. It shows that the structure of the language itself shapes how our thoughts work–everything from grammatical sentence structure to idioms. It lets a child realise that a foreign language is more than just an alien vocabulary being substituted in for our own words–and by the same token, that a foreign citizen is something other than just an American speaking a funny language and wearing funny clothes. They’re a product of their culture, shaped by that culture’s language and history and values–and hopefully, it also lets them realise that we have been wholly shaped by our own unique culture.

History. We’re taught that history is what’s happened in the past, but of course it’s not. History is what our sources tell us happened in the past. And that’s something that can be hard to understand–that everything we’re told in a history book is something that’s been filtered through someone else’s interpretation. History itself is interpretation–there’s no such thing as “objective history”. A student of history learns to question each piece of information they’re given–to examine it and to evaluate its source, to decide for themselves whether it should be trusted rather than simply accepting it without question.

And there we have it. There’s a lot of overlap in these skills, but that’s as it should be–they give our children a way of looking at the world. A way of seeing beneath the surface, of questioning everything, of examining their experiences and hopefully seeing through to what’s going on between the lines.

So … who can tell me what three things are defined by any set of three nonlinear points?

I

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Unto others

Peter Purves as Steven Taylor in 'The Celestial Toymaker'Lisa and I are big fans of Jeopardy!–big enough fans that we notice patterns in how the contestants play.

Ofttimes, if the contestant who is in the lead going into the Final Jeopardy round isn’t very confident about the Final Jeopardy category, they’ll base their wager on the amount of money that the second-place contestant has. That is, they’ll figure out how much the second-placed contestant can make by doubling their money, and they will wager just enough money to beat that total.

An example. Let’s say the first-placed contestant has eleven thousand dollars and the second-placed contestant has eight thousand dollars. The first-placed contestant therefore knows that the most money the second-placed contestant can earn is sixteen thousand dollars (by wagering all eight thousand dollars and getting the final question right). So the first-placed contestant needs to wager only five thousand dollars (16,000-11,000=5000) in order to guarantee either winning or tying (assuming they get the Final Jeopardy question correct) and coming back to play again tomorrow.

But I find that, overwhelmingly, in such a situation, the first-placed contestant doesn’t wager five thousand dollars. Instead, he or she wagers five thousand and one dollars. And I find this absolutely disgusting. If two (or all three, as has happened once in Jeopardy!’s history) contestants end up tied, both contestants win. Jeopardy! has no tiebreaker process. The leading contestant gains nothing at all by wagering that one extra dollar.

What’s more, that extra dollar actually harms the leading contestant. All these wagers are being made under the assumption that you are just as likely to get the Final Jeopardy question wrong and lose the money being wagered as you are to get the question right. If someone were to assume that they would definitely get the question right, they’d simply wager their entire total, whatever that number is. So when the first-placed contestant decides to wager only $5001 instead of wagering all eleven thousand dollars, they’re hedging their bets that they might get Final Jeopardy wrong–they’re wagering the minimum amount they need to guarantee victory, so that they’ll be left with the maximum remainder if they get the question wrong.

If they’d only wagered five thousand dollars, they’d be left with six thousand dollars if they got the question wrong. But by wagering $5001, they’ll only be left with $5999. They’re actually worsening their own position just so they can fuck over their fellow contestant.

Let’s say the second-placed contestant concludes that, if the first-placed contestant gets the question correct, then the second-placed wager won’t matter–the first-placed contestant will win anyway. So the second-placed contestant therefore decides to base their wager on the assumption that the first-placed contestant will get the question wrong, and wagers two thousand dollars–this will leave them six thousand dollars if they get the question wrong, the maximum amount that the first-placed contestant can also be left with if they get the question wrong.

(Please note that the second-placed contestant is being friendly, and not wagering only $1999. A wager of two thousand dollars leaves the way open for the coldhearted first-placed contestant to achieve a tie, whereas $1999–guaranteeing the second-placed contestant a minimum of $6001–would prevent that.)

The question is asked, the answers are given. The third-placed contestant’s answer doesn’t matter (I’m assuming here that the third-placed contestant has less than three thousand dollars). The second-placed contestant’s answer is wrong, leaving them with six thousand dollars. And then the first-placed contestant’s answer is also wrong, leaving them with $5999–meaning they lose by one dollar.

What gets me–what absolutely and truly disgusts me–is that the first-placed contestant on Jeopardy! will wager that extra dollar nine times out of every ten. For the overwhelming majority of people, it never seems to occur to them to not screw over their fellow contestant at no cost to themselves–even though that extra dollar actually has the potential to cost them the game.

I think, in all honesty, that says something rather ugly about us as a society.

I

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Based on previous experience

I spent election night watching the coverage on MSNBC. As it became more and more apparent that the Virginia Senate race was going to a recount (apparently, election-deciding recounts follow me around like the plague), one of their pundits mentioned that no one has ever won a recount without already having won the initial count (as, indeed, has now happened again).

Sooooo. If the original winner of the election has always won the recount … how come we need a recount at all, people?

In both Florida in 2000 and Virginia in 2006, the state recount law kicked in because the margin of victory was less than one half of one per cent. If you’re worrying about one half of one per cent of the votes being counted wrong, that means you’re worrying about a mistake being made with one out of every two hundred ballots cast. Quite apart from the new prevalence of electronic ballots–where it would be impossible for the computer to “recount” and get a different number than it had the first time–doesn’t that seem like an awfully high error rate to reasonably expect your people to be making? In an election like the one in Virginia, where 2.4 million people voted, that means you’re saying that you would be unsurprised if your electioneers miscounted twelve thousand ballots.

Maybe these were the same people who were in charge of purchasing enough guns and body armour for our troops in Iraq.

I

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Simple math

So yesterday I was watching the Cowboys-Redskins game about which I’m no doubt going to be subjected to conversation at work tonight. Early on, Dallas scored a touchdown to take the lead 6-5 and decided to go for two. Both commentators–Joe Buck and Troy Aikman–expressed a great deal of scepticism because it was just too early in the game to go for two. At halftime, Jimmy Johnson also condemned the decision–apparently, two-point conversions simply don’t produce enough points to merit going for them insteading of attempting a point-after until it’s absolutely necessary.

Except that when Dallas was lining up for their attempt, Joe Buck informed me that, “On two-point conversion attempts this season, NFL teams are fourteen for 23.”

Now, according to NFL.com, placekickers are 551 of 555 on point-after attempts so far this season, or about 99.3%. So I think that for the sake of argument, we can say that out of a given 23 PATs, placekickers are likely to convert 23 of them. That gives you 23 points.

But in the same number of two-point conversions, your offense can apparently get you twenty-eight points.

So–excepting late-game situations where you need a single point to tie or take the lead–why would teams ever go for a point after when they’d score more points by going for two?

It does occur to me that teams would get better at defending two-point attempts if they had to do so more often, but you don’t know how much better unless you try. And surely they’d get better at making the attempts as well.

I

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Dividing by 0

You know, sometimes mathematicians really stick in my craw. They blather on about how perfect and beautiful mathematics are, how they’re so ordered and logical and how everything just falls into place. But then they can’t come up with a nice, simple, elegant, logical answer for what–I should think–is probably one of the most obvious questions a budding amateur mathematician or layman can ask: What is the answer when you divide a number by zero? Dividing by zero, according to mathematicians, doesn’t have a “real” answer; x divided by zero is just “undefined”. And if you point out this glaring inconsistency to any of these number-lovin’ charlatans, are they at all apologetic for the gaping whole in their constructed vision of the universe? No! They look at you like you’re an illiterate–or, I guess, innumerate–moron not fit to do their simple addition for them, all because you can spot the obvious problem that is apparently beyond their comprehension.

Well this ends here. Right here, right now, we’re going to come up with the rule for dividing by zero. For this shouldn’t just be something you calculate out, the same way you would calculate out, say, 69 x 37. No, this is such a basic, obvious operation that, like dividing by one or by dividing a number by itself, it needs a rule. A theorem, perhaps.

My friend Nikki suggested that we create a new set of numbers for the operation of dividing by zero and assign them their own variable (perhaps z), much as the set of imaginary numbers, which are created when we take the square root of a negative number (another copout by these so-called mathematicians–whose calculations rapidly seem to be taking on all the credibility of alchemy), are assigned the variable i. But no–I cannot be fobbed off with such an easy out; my standards of mathematics are too high, too pure. I want a number, damn it, not any of these stupid letternumbers the more pedestrian mathematicians are so proud of.

And that number is–0. X over 0 shall from now on equal 0, just as x over 1 equals x and x over x equals 1. I give you, ladies and gentlemen, the PythagorIan Theorem. And also, I give you its sole exception–for unlike the number lovers, who start shrieking in fear once you ask, “Where is the exception that proves this rule, for without an exception it has not been proved?”, I who minored in English fully understand the vital necessity of the exception that proves the rule. And the exception that proves the PythagorIan Theorem is that 0 divided by 0–being the same as x over x–equals 1.

I’m glad we’ve finally laid to rest this most ancient of mathematical controversies. Ladies and gentlemen, we have made history here today.

I

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